| L(s) = 1 | + 3i·3-s + 4.43i·7-s − 9·9-s + 3.43·11-s − 78.7i·13-s + 53.1i·17-s + 20.4·19-s − 13.3·21-s + 118. i·23-s − 27i·27-s − 168.·29-s + 61.0·31-s + 10.3i·33-s − 246. i·37-s + 236.·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.239i·7-s − 0.333·9-s + 0.0941·11-s − 1.67i·13-s + 0.758i·17-s + 0.246·19-s − 0.138·21-s + 1.07i·23-s − 0.192i·27-s − 1.07·29-s + 0.353·31-s + 0.0543i·33-s − 1.09i·37-s + 0.969·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.430317919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.430317919\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.43iT - 343T^{2} \) |
| 11 | \( 1 - 3.43T + 1.33e3T^{2} \) |
| 13 | \( 1 + 78.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 53.1iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 20.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 118. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 168.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 61.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 246. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 422.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 362. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 170. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 546. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 216.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 130.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 614. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 324.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 88.8iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 758. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 195.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 521iT - 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.606169706994103142619356889603, −8.959196014695245875718079127102, −7.950868239527876119331997786800, −7.40990365039025736032497137631, −5.85256095244424261313719914745, −5.66942260432233953726970742188, −4.41729811100287958464120327110, −3.49606943700299413642403387292, −2.58677224690116237412795613278, −1.08469285403054432347843573352,
0.37233289806327344204094017498, 1.64402485410868897552531765326, 2.60953511303740975995487604555, 3.90318851727986959457799977599, 4.75953747566666428834963987920, 5.86897009901234364799008531158, 6.82033605566956695935352722970, 7.23571242924097352474850597317, 8.318324401998223680088563120590, 9.084806307510191194117282177691
